Remark 4 From Theorem REF , we can explicitly construct a new family of Singleton-optimal locally repairable codes with length up to \(q-1+\lceil \frac{q-1}{r}\rceil \) , which is larger than \(q\)  compared with [1]}, [2]}, [3]}. If the minimum distance \(d\)  is large and linearly proportional to \(q\) , then \(r\)  is linear proportional to \(q-1\)  as well, i.e., \(\lceil \frac{q-1}{r}\rceil \)  is a constant with respect to large \(q\) .
In this case, the length of such Singleton-optimal locally repairable codes is less than the \(q+2\sqrt{q}\)  which can be obtained from elliptic curves [4]}, [5]}.

